Apollonius <Pergaeus>, Apollonii Pergaei Conicorvm Lib. V. VI. VII. paraphraste Abalphato Asphahanensi : nunc primum editi ; additvs in calce Archimedis assvmptorvm liber, ex codibvs arabicis mss Abrahamus Ecchellensis Maronita latinos reddidit, Jo. Alfonsvs Borellvs curam in geometricis versione contulit & [et] notas vberiores in vniuersum opus adiecit

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        <div xml:id="echoid-div633" type="section" level="1" n="210">
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            <s xml:id="echoid-s6715" xml:space="preserve">
              <pb o="175" file="0213" n="213" rhead="Conicor. Lib. VI."/>
            mus, &</s>
            <s xml:id="echoid-s6716" xml:space="preserve">c. </s>
            <s xml:id="echoid-s6717" xml:space="preserve">Expoſitio, atque demonſtratio huius propoſitionis obſcura eſt propter
              <lb/>
            nimiam eius breuitatem: </s>
            <s xml:id="echoid-s6718" xml:space="preserve">itaque duo eius caſus diſtingui debent hac ratione. </s>
            <s xml:id="echoid-s6719" xml:space="preserve">In
              <lb/>
            duobus triangulis A B C, D E F ſupponantur anguli H, & </s>
            <s xml:id="echoid-s6720" xml:space="preserve">I æquales, pariter-
              <lb/>
            que anguli H B A, I E D æquales inter ſe; </s>
            <s xml:id="echoid-s6721" xml:space="preserve">ideoque duo triangula A B H, & </s>
            <s xml:id="echoid-s6722" xml:space="preserve">
              <lb/>
            D E I ſimilia erunt, & </s>
            <s xml:id="echoid-s6723" xml:space="preserve">propterea A H ad H B eandem proportionem habebit,
              <lb/>
            quàm D I ad I E; </s>
            <s xml:id="echoid-s6724" xml:space="preserve">ſed ex vniuerſali hypotheſi rectangulum C A H ad quadra-
              <lb/>
            tum H B eandem proportionem habet, quãm rectangulum F I D ad quadratum
              <lb/>
            I E, & </s>
            <s xml:id="echoid-s6725" xml:space="preserve">componuntur proportiones rectangulorum ad quadrata iam dicta ex ra-
              <lb/>
            tionibus laterum circa angulos æquales H, & </s>
            <s xml:id="echoid-s6726" xml:space="preserve">I, ſuntque oſtenſæ proportiones A
              <lb/>
            H ad H B, atque D I ad I E eædem inter ſe; </s>
            <s xml:id="echoid-s6727" xml:space="preserve">igitur reliquæ componentes pro-
              <lb/>
            portiones, ſcilicet C H ad H B, atque F I ad I E eædem quoque erunt inter ſe,
              <lb/>
            & </s>
            <s xml:id="echoid-s6728" xml:space="preserve">compræhendunt angulos æquales H, & </s>
            <s xml:id="echoid-s6729" xml:space="preserve">I; </s>
            <s xml:id="echoid-s6730" xml:space="preserve">igitur triangula C H B, & </s>
            <s xml:id="echoid-s6731" xml:space="preserve">F I E
              <lb/>
            ſimilia ſunt inter ſe: </s>
            <s xml:id="echoid-s6732" xml:space="preserve">& </s>
            <s xml:id="echoid-s6733" xml:space="preserve">propterea angulus B C A æqualis erit angulo E F D,
              <lb/>
            ſed anguli B A C, & </s>
            <s xml:id="echoid-s6734" xml:space="preserve">E D F æquales ſunt inter ſe, quia eorum conſequentes
              <lb/>
            æquales erant in triangulis æquiangulis B A H, & </s>
            <s xml:id="echoid-s6735" xml:space="preserve">E D I, igitur duo triangu-
              <lb/>
            la B A C, & </s>
            <s xml:id="echoid-s6736" xml:space="preserve">E D F æquiangula, & </s>
            <s xml:id="echoid-s6737" xml:space="preserve">ſimilia inter ſe erunt.</s>
            <s xml:id="echoid-s6738" xml:space="preserve"/>
          </p>
          <p style="it">
            <s xml:id="echoid-s6739" xml:space="preserve">Simili modo ſi ſupponantur anguli C B H, & </s>
            <s xml:id="echoid-s6740" xml:space="preserve">F E I æquales, cum anguli H,
              <lb/>
            & </s>
            <s xml:id="echoid-s6741" xml:space="preserve">I æquales ſint, erunt triangula B C H, & </s>
            <s xml:id="echoid-s6742" xml:space="preserve">E F I ſimilia inter ſe, & </s>
            <s xml:id="echoid-s6743" xml:space="preserve">vt prius,
              <lb/>
            oſtendentur quoque triangula ablata B A H, E D I æquiangula, & </s>
            <s xml:id="echoid-s6744" xml:space="preserve">ſimilia in-
              <lb/>
            ter ſe (propterea quod circa angulos æquales H, & </s>
            <s xml:id="echoid-s6745" xml:space="preserve">I babent latera proportiona-
              <lb/>
            lia); </s>
            <s xml:id="echoid-s6746" xml:space="preserve">& </s>
            <s xml:id="echoid-s6747" xml:space="preserve">ideo reſidua triangula C A B, & </s>
            <s xml:id="echoid-s6748" xml:space="preserve">F D E erunt quoque ſimilia, vt
              <lb/>
            propoſitum fuerat.</s>
            <s xml:id="echoid-s6749" xml:space="preserve"/>
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        <div xml:id="echoid-div635" type="section" level="1" n="211">
          <head xml:id="echoid-head266" xml:space="preserve">SECTIO SEXTA</head>
          <head xml:id="echoid-head267" xml:space="preserve">Continens Propoſit. XV. XVI. & XVII.</head>
          <head xml:id="echoid-head268" xml:space="preserve">PROPOSITIO XV.</head>
          <p>
            <s xml:id="echoid-s6750" xml:space="preserve">DVarum hyperbolarum, aut ellipſium, ſi figuræ diametro-
              <lb/>
            rum, quæ axes non ſint, fuerint ſimiles, atque potentes
              <lb/>
            contineant cum diametris angulos æquales: </s>
            <s xml:id="echoid-s6751" xml:space="preserve">vtique ſectiones
              <lb/>
            ſunt ſimiles.</s>
            <s xml:id="echoid-s6752" xml:space="preserve"/>
          </p>
          <p>
            <s xml:id="echoid-s6753" xml:space="preserve">Sint ſectiones A B, C D hyperbolicæ, vel ellipticæ earum diametri,
              <lb/>
            quæ non ſint axes I A K, L C M, & </s>
            <s xml:id="echoid-s6754" xml:space="preserve">earum centra G, H, & </s>
            <s xml:id="echoid-s6755" xml:space="preserve">duo axes
              <lb/>
            ſint E B, F D: </s>
            <s xml:id="echoid-s6756" xml:space="preserve">& </s>
            <s xml:id="echoid-s6757" xml:space="preserve">educamus duas tangentes A R, C S ad duos axes,
              <lb/>
            quæ continebunt cum duabus diametris A K, C M duos angulos æqua-
              <lb/>
            les, eo quod parallelæ ſunt potentialibus ad diametros eductis; </s>
            <s xml:id="echoid-s6758" xml:space="preserve">& </s>
            <s xml:id="echoid-s6759" xml:space="preserve">edu-
              <lb/>
            camus à B, D ad duabus diametros A K, C M tangentes B N, D O, & </s>
            <s xml:id="echoid-s6760" xml:space="preserve">
              <lb/>
            circumducamus ſuper triangula B N G, H D O duos circulos, & </s>
            <s xml:id="echoid-s6761" xml:space="preserve">ex A,
              <lb/>
            C educamus ad axes duas potentiales A P, C Q, & </s>
            <s xml:id="echoid-s6762" xml:space="preserve">per B, D ducamus
              <lb/>
            I B T, L D V parallelas ipſis A R, C S, quæ ſecent duos circulos in B,
              <lb/>
            T, D, V: </s>
            <s xml:id="echoid-s6763" xml:space="preserve">eritque G I in I N, ſcilicet ei æquale T I in I B ad quadra-
              <lb/>
              <note position="left" xlink:label="note-0213-01" xlink:href="note-0213-01a" xml:space="preserve">b</note>
            </s>
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